The Shakura-Sunyaev viscosity prescription with variable (r)

Abstract

Almost all hydrodynamic accretion disc models parametrize viscosity with the dimensionless parameter α. There is no detailed model for α, so it is usually taken to be a constant. However, global simulations of magnetohydrodynamic discs find that α varies with distance from the central object. Also, Newtonian simulations tend to find smaller α’s than general relativistic simulations. We seek a one-dimensional model for α that can reproduce these two observations. We are guided by data from six general relativistic magnetohydrodynamic accretion disc simulations. The variation of α in the inner, laminar regions of the flow results from stretching of mean magnetic field lines by the flow. The variation of α in the outer, turbulent regions results from the dependence of the magnetorotational instability on the dimensionless shear rate. We give a one-dimensional prescription for α(r) that captures these two effects and reproduces the radial variation of α observed in the simulations. For thin discs, the prescription simplifies to the formula α(r) = 0.025[q(r)/1.5]6, where the shear parameter, q(r), is an analytical function of radius in the Kerr metric. The coefficient and exponent are inferred from our simulations and will change as better simulation data becomes available. We conclude that the α-viscosity prescription can be extended to the radially varying α’s observed in simulations. It is possible that Newtonian simulations find smaller α’s than general relativistic simulations because the shear parameter is lower in Newtonian flows.